Korea-Australia Rheology Journal Vol. 21, No. 2, June 2009 pp. 119-126 Numerical investigation on the blood flow characteristics considering the axial rotation in stenosed artery Kun Hyuk Sung, Kyoung Chul Ro and Hong Sun Ryou* School of Mechanical Engineering, Chung-Ang University, 221 HeukSuk-Dong, Dongjak-Gu, Seoul 156-756, Korea (Received March 13, 2009; final revision received May 4, 2009) Abstract A numerical analysis is performed to investigate the effect of rotation on the blood flow characteristics with four different angular velocities. The artery has a cylindrical shape with 50% stenosis rate symmetrically distributed at the middle. Blood flow is considered a non-newtonian fluid. Using the Carreau model, we apply the pulsatile velocity profile at the inlet boundary. The period of the heart beat is one second. In comparison with no-rotation case, the flow recirculation zone (FRZ) contracts and its duration is reduced in axially rotating artery. Also wall shear stress is larger after the FRZ disappears. Although the geometry of artery is axisymmetry, the spiral wave and asymmetric flow occur clearly at the small rotation rate. It is caused that the flow is influenced by the effects of the rotation and the stenosis at same time. keywords : blood flow, axially rotating velocity, stenosis, pulsatile flow, non-newtoninan 1. Introduction Arteriosclerosis is one of the most widespread diseases in human beings. It is a significant factor in the death rate because it affects hemodynamics and reduces the flow rate of blood to the heart and the brain. In particular, sudden body movement causes the blood pressure fluctuations of a person with an arterial diseases and an ischemic poverty of blood to increase more than those of a healthy person. In a serious case, when the blood flow rate decreases, the person may fall because of vertigo and experience temporary eyesight and hearing trouble. Thus, human body movements can affect the blood flow characteristics of persons with arterial diseases more than those of healthy persons. In research on human acceleration, Burton et al. (1974) studied eyesight trouble caused by a change of blood flow in extreme gravity circumstances. Hooks et al. (1972) conducted a clinical study of the side effects of body acceleration. In research on blood flow and acceleration, Misra and Sahu (1988) developed a mathematical model to study the blood flow through large arteries under the action of periodic body acceleration. Belardinelli et al. (1989) performed an experimental study of the effect of blood pressure by shock acceleration, and Mandal et al. (2007) performed a numerical study on the blood characteristics of a cylindrical blood vessel with periodic accelerations. Nakamura et al. (1988) and Luo et al. (1992) carried out an investigation *Corresponding author: cfdmec@cau.ac.kr 2009 by The Korean Society of Rheology of the blood flow characteristics of stenosed and bifurcated blood vessels. Ro et al. (2008) performed a numerical study on the blood characteristic of the carotid bifurcation artery with periodic accelerations. However, none of these researches focused on the effect of body acceleration on the characteristics of blood flow without rotation. From the medical side, studies on the effects of cervical or spinal rotation on hemodynamic in arteries have been executed, but they concentrated on the velocity distribution caused by the change in artery volume. There have been only a few studies on how the rotational movement of the human body affects blood flow characteristics. In fluid dynamics, the flow characteristics of an axially rotating pipe without stenosis have been studied. For example, Imao et al. (1992) showed the flow instability problem in axially rotating pipes at the critical ratio of the circumferential velocity to the mean axial velocity. Kikuyama et al. (1983) showed that the transition of flow state from the laminar to the turbulent in axially rotating pipe can occur at a low Reynolds number. Both of these experiments showed that rotation caused destabilization of the flow in an axially rotating pipe. However, these experiments used water, so they did not address the characteristics of blood, which has non-newtonian viscosity. Generally, the assumption of Newtonian behavior of blood is acceptable for high shear-rate flow, but it is not valid when the shear rate is low (0.1 s -1 ), as it is in small arteries or on the downstream side of stenosis (Chien et al., 1982). It has also been pointed out that, in some diseased conditions e.g. patients with severe myocardial infarction, cerebrovascular diseases Korea-Australia Rheology Journal June 2009 Vol. 21, No. 2 119
Kun Hyuk Sung, Kyoung Chul Ro and Hong Sun Ryou and hypertension blood exhibits remarkable non-newtonian behavior. Thus, non-newtonian viscosity must be considered when analyzing the characteristics of flow in stenosed arteries. Young (1979) showed that head loss is a nonlinear function of stenosis and that pressure losses become significant only for stenoses greater than 50 ~70%. Thus, the stenosis affects significantly the pressure distribution in the artery and puts the person with an arterial disease in jeopardy. The experiment showed the effect of inlet velocity on the flow characteristics at the downstream of stenosis in the artery (Deplano et al., 1999). Extremely, the angular velocity is about 8 revolutions a second when the figure skater performs a standing spin. Although that case is rare to the common people, they undergo rotations of body through daily exercise. So rotation of body can affect the characteristic of blood flow such as pressure drop, wall shear stress, and flow recirculation zone. However, in spite of the importance of rotation effect, there is no research on the effect of rotation on the flow characteristics considered pulsatile velocity profile in stenosed arteries. Hence, for the basic study on the rotation effect to the blood flow, we select the common carotid artery because the artery rotates axially when people do a standing spin. Therefore the purpose of this paper is a numerical analysis of the effect of rotation and an unsteady pulsatile velocity profile in a stenosed artery. 2. Numerical details 2.1. Governing equations In order to simulate the blood flow characteristics, mass and momentum conservation equations are required and the non-newtonian viscosity and pulsatile flow must be considered. In addition, the axially rotating, centrifugal force must be added to the momentum equation as a source term. ρ ----- + ( ρν) = 0 t --- ( ρν) + ( ρν ν) = t ( pδ + µ ( ν+ ( ν)t) ) ρω ( ω r) (2) To simulate a non-newtonian fluid problem, a constitution equation is required for blood rheology characteristics described by the second invariant of shear rate tensor: τ= ηγ (3) where η and γ are apparent viscosity and shear rate. Shear rate γ is represented as: γ 1 = -- γ ijγji 2 (4) i j (1) Fig. 1. The schematic view and grid generation of stenosed blood vessel. We use the Carreau viscosity model because it is more suitable for representing blood rheology characteristics (Cho, 1985): η = η + ( η 0 η )[ 1 + ( λγ ) 2 ] ( n 1) 2 (5) where η 0 is the zero shear viscosity (0.056 Pa s), η is the infinite shear viscosity (0.00345 Pa s), λ is the time constant (3.313 s) and n is the power law index (0.356). 2.2. Modeling of an artery with stenosis Fig. 1 shows a schematic view and grid generation of a stenosed blood vessel. The stenosis, where is located between 1D upstream and 1D downstream from the center of stenosis, is modeled by Young s model (Young, 1968), as shown in equation (6). The diameter of the blood vessel is 8 mm, and the minimum diameter of the stenosis is half the size of the blood vessel. The stenosis rate is 50% with no eccentricity. Rz () = [ R a R[ 1 + cosπ( z z 1 ) z 0 ]] (6) where, a is the stenosis rate, z 0 is half the length of the stenosis,1d, and z 1 is the axial position from the starting point of the stenosis where is 1D upstream from the center of stenosis. The grid independent test is performed with four different number of grid cells which are 43,587, 130,720, 434,808 and 580,320 when the angular velocity is 6 rev/s. In Fig. 2(a) and (b), the averaged wall shear stress (WSS) with 434,808 grid cells follows that with 580,320 grid cells at each 3D and 5D downstream from the center of stenosis. The difference of averaged WSS is within 5% for a period of pulsatile in both cases. Thus, 434,808 hexahedral grid cells is selected for numerical analysis. The computing time for each case was about 8 hours with 8 nodes, 2.0 GHz CPU. 2.3. Boundary and initial condition For the numerical simulation, the 3-D time-dependent Navier-Stokes equations were solved by the ANSYS CFX V11.0 based on the finite volume method with the pressure-based coupled solver. Fixed time step, 0.002 s, was used with the Second Order Backward Euler scheme for transient term. The flow is assumed to be a laminar flow, incompressible, non-newtonian, and the wall of artery is 120 Korea-Australia Rheology Journal
Numerical investigation on the blood flow characteristics considering the axial rotation in stenosed artery Fig. 3. The pulsatile inlet velocity profile. rigid with no slip conditions. For the study of the effect of the pulsatile flow on the blood flow characteristics in an artery, the idealized pulsatile velocity profile of the common carotid artery is used as the inlet boundary condition (Gijsen et al., 1999). Fig. 3 shows the pulsatile velocity profile which is dimensionless by the period of pulsatile cycle, t p, which is 1 s. The timeaveraged dynamic viscosity for a period of pulsatile is 0.007865 Pa s. The peak of velocity is 0.20 m/s at inlet in pulsatile and the density of blood is 1100 kg/m 3. The pressure boundary condition is used in the outlet of the artery. The initial velocity through whole domain is equal to the start of systole, t/t p =0, in pulsatile. For the study on the effect of rotation on the blood flow characteristics, we use a MRF (Multiple Reference Frame) method for application to the rotating effect of blood vessel (Luo et al., 1994). Through the MRF method, the only wall is rotated. 3. Results and discussion Fig. 2. Averaged axial WSS on circumferential lines at two different locations from the stenosis in pulsatile for various number of grid cells. ((a) At 3D downstream from the stenosis, (b) At 5D downstream from the stenosis). For the validation of our numerical method, the numerical results are compared with the experiment in axially rotating pipe (Imao et al., 1992). The angular velocity of an axially rotating artery is varied as 1, 2, 4 and 6 revolutions per a second (rev/s), and the results are compared with those of the no-rotation case. We compare the axial velocity profiles of Newtonian fluid flow to that of non-newtonian fluid flow for 6 rev/s. Consequently, the Newtonian fluid flow is more unstable than another due to the magnitude of viscosity. Overall, the viscosity of non-newtonian fluid is larger than that of Newtonian fluid. For unsteady flow, the simulations are executed over at least three cycles to achieve a periodic solution. The velocity variation after two cycles is less than 1% at test points behind the stenosis and results are saved for the final cycle. In this section, the blood flow characteristics such as axial velocity profiles, pressure distribution, flow recirculation zone (FRZ) and wall shear stress (WSS) distribution are presented. The blood flow characteristics are obtained for the entire flow domain at four different instants (t / t p = 0.14, 0.16, 0.40, 0.78) in pulsatile. 3.1. Validations Due to the instability problem of flow in the experiment Korea-Australia Rheology Journal June 2009 Vol. 21, No. 2 121
Kun Hyuk Sung, Kyoung Chul Ro and Hong Sun Ryou Fig. 5. Axial velocity profiles and flow recirculation zone at different time phase in pulsatile. The position of profile is presented as the multiple of diameter distal to stenosis. ((a) No rotation model, t/t p =0.14, (b) No rotation model, t/t p =0.16, (c) No rotation model, t/t p =0.40, (d) No rotation model, t/t p =0.78, (e) 6 rev/s model, t/t p =0.40, (f) 6 rev/s model, t/t p =0.78). The numerical results follow the experimental results except the axial velocity profile at Z = 120. The profile of measured axial velocity tends to become the turbulent at Z = 120. This reason is that the flow state changes from laminar to turbulent by the rotation effect in experiment. However, the circumferential velocity profile approaches the solid-body rotation gradually as the numerical simulation predicts. Fig. 4. Dimensionless velocity profile ((a) The dimesionless circumferential velocity, (b) The dimensionless axial velocity). (Imao et al., 1992), the spiral wave appears as the ratio of circumferential velocity to the axial velocity increases and it is clear when the rotation rate is 3 in axially rotating pipe. Thus, we have been performed the numerical analysis when the rotation rate is 3. In the experiment, the Reynolds number is 500 and the ratio of circumferential velocity to the axial velocity is 3. The circumferential velocity of a pipe and the flow rate of water are given as constant. Fig. 4(a) shows the circumferential velocity which is dimensionless by the wall velocity and Fig. 4(b) shows the axial velocity which is dimensionless by the inlet velocity in axially rotating pipe. 3.2. Results of numerical analysis The axial velocity profiles are presented in a plane containing the axis of artery because the geometry is axisymmetric. Fig. 5(a), (b), (c) and (d) show axial velocity profiles and FRZs when the artery is not rotating at different time phases in pulsatile. The contour of zero velocity indicates the boundary of FRZ, in which the flow is either stagnant or reversed. The jet velocity profile like a piston shape appears behind the stenosis in flow acceleration phase. Due to the decrease of the area at the stenosis and reverse flow near the vessel wall, axial velocity of the center line of artery increases in order to satisfy the flow rate conservation law at 1D downstream from the stenosis. The FRZ appears at all time in pulsatile because the stenosis causes disturbance to the flow and its size varies due to the change 122 Korea-Australia Rheology Journal
Numerical investigation on the blood flow characteristics considering the axial rotation in stenosed artery Fig. 7. The Axial velocity contours ((a) Axial velocity contours when the angular velocity is 6 rev/s at four different time phase in pulsatile, (b) Axial velocity contours as variation on the angular velocity at t/tp=0.40). Fig. 6. Pressure distribution at t/tp=0.40 ((a) Pressure distribution along the axial line at t/tp=0.40, (b) Pressure contour at yzplane as x=0). of magnitude of velocity in pulsatile. The FRZ grows and expands to the downstream of stenosis as flow accelerates and extends up to the whole downstream at t/tp =0.4. After that time, FRZ contracts rapidly and its size becomes constant because the gradient of velocity is zero in pulsatile. Due to the centrifugal force by rotating of vessel, the pressure along axial line decreases but the pressure near wall increases at the same time as shown in Fig. 6(a). Consequently, the iso-pressure lines become more convex as the angular velocity increases like Fig. 6(b). This causes the friction increases and this phenomenon suppresses the heart due to the increase of pressure drop at the rotating artery. Fig. 5(e) and (f) show axial velocity profiles and FRZs when the angular velocity is 6 rev/s. In comparison with the no-rotation case, the FRZs are reduced remarkably due to the increase of radial velocity from the centrifugal force at t/tp = 0.4. Also, axial velocity profiles and FRZs are asymmetry although the geometry of artery is axisymmetry. Fig. 7(a) shows the axial velocity contours when the angular velocity is 6 rev/s at four different time phase in pulsatile. Asymmetric contour with three or four protrusions appears at t/tp =0.4 and possesses the eccentricity. In Fig. 6, the protrusions constitute the spiral wave as the angular velocity is 4 rev/s. Fig. 7(b) shows axial velocity Korea-Australia Rheology Journal contours as variation on the angular velocity at t/tp =0.40. As the fluid flows to the downsteram, the four protrusions of contour become dim at 6 rev/s. On the other hand, the shape of four protrusions is maintained at 4 rev/s. The asymmetric coutours appear after 3D downstream from the stenosis at the whole rotating cases and after that location, 3D, the shape of protrusions rotates slowly. Imao et al. (1992) investigated the structure and characteristic of a spiral wave with the flow visualization technique in axially rotating pipe. In experimental study, the spiral wave occurs as the variation of rotation rate which is the ratio of circumferential velocity to the axial velocity. As the rotation rate is 3, the spiral wave is most amplified and is reduced with a greater rotation rate. Fig. 8 shows the Iso-contour of the axial velocity at four different instants in pulsatile. As the angular velocity is 4 rev/s, the spiral wave occurs at t/tp =0.4. The spiral wave appears and disappears as the variation of rotation rate due to the pulsatile inlet velocity. The stenosis affects the stability of flow in the artery. Tang et al. (1999) showed the asymmetric flow patterns occur and become unstable in an axisymmetric geometry with the stenosis. Buchanan et al. (1998) showed in their numerical study that two co-rotating vortices occurred in a 75% (area reduction) axisymmetrical stenosed model when the flow started to decelerate in pulsatile. In our numerical analysis, the spiral wave and asymmetric flow occur clearly although the rotation rate is smaller than 3. It is caused by the flow is influenced by June 2009 Vol. 21, No. 2 123
Kun Hyuk Sung, Kyoung Chul Ro and Hong Sun Ryou Fig. 8. ISO-contour of the axial velocity which is 0.065 m/s when the angular velocity is 4 rev/s. the effects of the rotation and the stenosis at the same time. Variations of averaged axial WSS on circumferential lines are shown in Fig. 9 at four different downstream locations through the whole pulsatile. Considering for the asymmetric flow due to the rotation and stenosis of artery, WSS are averaged on circumferential line of each location. The intersections of WSS curves and the horizontal axis are two points at which WSS changes signs and hence correspond to the flow separation point (WSS changes from positive to negative) and reattachment point (WSS changes from negative to positive). At 1D downstream from the stenosis, there is no intersection point and the sign of WSS is negative all the time. Those indicate FRZ occurs through the whole cycle. After 3D downstream from the stenosis, it is notable that the difference of negative WSS caused by the reverse flow. The WSS of no-rotation case is about 40% and nearly twice smaller than that of rotating case due to the stronger reverse flow at 5D and 7D downstream from the center of stenosis, respectively. The centrifugal force caused by rotation effect suppresses the blood flow toward wall and this pheonomenon decreases the reverse Fig. 9. Averaged axial WSS on circumferential lines at four different downstream locations in pulsatile ((a) 1D downstream from the stenosis, (b) 3D downstream from the stenosis, (c) 5D downstream from the stenosis, (d) 7D downstream from the stenosis). 124 Korea-Australia Rheology Journal
Numerical investigation on the blood flow characteristics considering the axial rotation in stenosed artery flow and the FRZs. But, the WSS of rotation is larger after the FRZs disappear because the circumferential gradient of velocity is producted by the rotation of artery. This causes the new vascular disorder by injuring endothelium of artery such as arteroscelrosis at the downstream of stenosis (Fry, 1972). Also, when the artery is rotating, the existence time of FRZs shortens further from the stenosis. In comparison with no-rotation case, the existence time is smaller about 12.5% at 7D downstream from the center of stenosis. 4. Conclusion In this paper, the rotation effects has been studied numerically on a stenosed blood vessel with the pulsatile inlet velocity profile. The FRZs occur through the whole pulsatile because the stenosis causes disturbance to the flow and its size varies in pulsatile. In comparison with the no-rotation case, the size and existence time of FRZs are reduced remarkably due to the increase of radial velocity from the centrifugal force in axially rotating artery. But, the friction increases due to the increase of pressure by the centrifugal force. This phenomenon suppresses the heart due to the increase of pressure drop at the rotating artery. Also, the WSS of no-rotation case is about 40% and nearly twice smaller than that of rotating case due to the stronger reverse flow at 5D and 7D downstream from the stenosis, respectively. But the WSS of rotation is larger after the FRZs disappear because the circumferential gradient of velocity is producted by the rotation of artery. This causes the new vascular disorder by injuring endothelium of artery such as arteroscelrosis at the downstream of stenosis. Although the geometry of artery is axisymmetry, the spiral wave and asymmetric flow occur clearly in spite of the small rotation rate. The contour of axial velocity is asymmetric with four or three protrusions at t/t p =0.4 and possesses the eccentricity after 3D downstream from the stenosis at the whole rotating cases. And it is caused that the flow is influenced by the effects of the rotation and the stenosis at the same time. Acknowledgement This research was partially supported by the Chung-Ang University Grant in 2009. List of symbols R [m] radius of blood vessel D [m] the maximum artery diameter t [s] simulation time t p [s] a period of pulsatile z 0 [m] half the length of the entire stenosis region, 1D z 1 [m] the axial position from the starting point of the stenosis region a [-] the rate of stenosis v [-] dimensionless circumferential velocity, v /R ω r [-] dimensionless radial distance, r /R w [-] dimensionless axial velocity ρ [kg/m 3 ] density ν [m/s] blood velocity tensor η [Pa s] apparent viscosity γ [1/s] shear rate ω [rev/s] angular velocity Subscripts 0 zero shear, amplitude infinite shear Superscripts ( ) Dimension value References Belardinelli, E., M. Ursino and E. Iemmi, 1989, A preliminary theoretical study of arterial pressure perturbations under shock acceleration, J. Biomech. Eng.-Trans. ASME 111, 233-240. Buchanan Jr, J. and C. Kleinstreuer, 1998, Simulation of particlehemodynamics in a partially occluded artery segment with implications to the initiation of microemboli and secondary stenoses, J. Biomech. Eng.-Trans. ASME 120, 446-454. Burton, R. R., S. D. Leverett Jr and E. D. Michaelson, 1974, Man at high sustained +G(z) acceleration: a review, AEROSPACE MED. 45, 1115-1136. Chien, S., 1982, Hemorheology in clinical medicine, Clin. Hemorheol. 2, 137-142. Cho, Y. I., L. H. Back and D. W. Crawford, 1985, Experimental investigation of branch flow ratio, angle, and Reynolds number effects on the pressure and flow fields in arterial branch models, J. Biomech. Eng.-Trans. ASME 107, 257-267. Deplano, V. and M. Siouffi, 1999, Experimental and numerical study of pulsatile flows through stenosis: Wall shear stress analysis, J. Biomech. 32, 1081-1090. Fry, D., 1972, Response of the arterial wall to certain physical factors. Atherogenesis: Initiating factors, A Ciba Foundation Symp., 40-43. Gijsen, F. J. H., E. Allanic, F. N. Van De Vosse and J. D. Janssen, 1999, The influence of the non-newtonian properties of blood on the flow in large arteries: Unsteady flow in a 90 curved tube, J. Biomech. 32, 705-713. Hooks, L., R. Nerem and T. Benson, 1972, A momentum integral solution for pulsatile flow in a rigid tube with and without longitudinal vibration, Int. J. Eng. Sci. 10, 989-1007. Imao, S., M. Itoh, Y. Yamada and Q. Zhang, 1992, The characteristics of spiral waves in an axially rotating pipe, Exp. Fluids Korea-Australia Rheology Journal June 2009 Vol. 21, No. 2 125
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